Metrization in small and large scale structures

Takahisa Miyata¹

¹ Department of Mathematics and Informatics Graduate School of Human Development and Environment Kobe University Kobe, 657-8501 Japan

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 65 (2017) no. 1, pp. 81-92

Résumé

Given a topological structure and a coarse structure on a set, N. Wright gave a necessary and sufficient condition for the set to have a metric inducing simultaneously both the structures. We use the idea of the Alexandroff and Urysohn metrization theorem for topological spaces, to investigate a simultaneous metrization problem for a set with a uniform (and topological) structure and a coarse structure. In particular, we prove that given two metrics $d_U$ and $d_C$ on a set $X$ such that the uniform (topological) structure induced by $d_U$ is compatible in some sense with the coarse structure induced by $d_C$, there exists a metric $d$ on $X$ which is isometric to $d_U$ in a small scale and to $d_C$ in a large scale. We then apply this idea to show that if, in addition, the uniform space has uniform dimension $0$ and the coarse space has asymptotic dimension $0$, then there exists an ultrametric $d$ on $X$ which is isometric to $d_U$ in small scale and to $d_C$ in large scale.

Détail

DOI : 10.4064/ba8081-4-2017

Keywords: given topological structure coarse structure set wright gave necessary sufficient condition set have metric inducing simultaneously structures idea alexandroff urysohn metrization theorem topological spaces investigate simultaneous metrization problem set uniform topological structure coarse structure particular prove given metrics set uniform topological structure induced compatible sense coarse structure induced there exists metric which isometric small scale large scale apply idea addition uniform space has uniform dimension coarse space has asymptotic dimension there exists ultrametric which isometric small scale large scale